In English it's possible to say last last year to informally say "the year before the last". Is it similar in Japanese that I can say きょうきょう年 to mean the same?
Last year (きょねん) is typically written as 去年.
While the word 去去年 (きょきょねん) seems to exist in the dictionary, I've never heard this usage before.
The phrase 一昨年 (おととし) is used more commonly for "the year before last".
EDIT: A few searches come up mostly with Chinese results, confirming 去去年 isn't common in Japanese.
Be aware! If you ask somebody "What did you do KYO-O-NEN - 享年? , he'll be startled out. Bcause you are asking him "What did you do in the year you were dead?
去年 (きょねん）means "last year," but 享年（きょうねん）means "the year somebody died," which is inscribed on the back of a tomb stone under the name of the dead.
There is no such a word as 去去年 in Japanese language. Perhaps you are refering to "一昨年" - the year before the last year -, or two years ago, which is pronounced as "Issakunen" or "Ototosi."